<< EDCRGTCcode.m

Spacetime: CONFORMAL TO Schwarzschild with Cosmological Constant and time-dependent mass (M[t])
	(All timings refer to an Apple iBook G3/900 computer running Mathematica 3.0 under Mac OS 9.2.2)

xCoord = {t, r, θ, φ};

g = Ω[t, r]^2 
           DiagonalMatrix[{1 - 
                2  M[t]/r - Λ r^2/3, -1/(1 - 
                    2  M[t]/r - Λ r^2/
                        3), -r^2, -r^2 
                Sin[θ]^2}];

simpRules = TrigRules;

RGtensors[g, xCoord]

•  Sample components:

Rdd[[3, 3]]

R

Wdddd[[1, 2, 1, 2]]

• Bach Tensor definition  \(W\_\(\(\(a\)\(\ \ \)\(b\)\(\ \ \ \)\)\(;\)\(q\)\(;\)\(p\)\(\ \)\)\%\(\(\ \ \)\(p\ \ q\)\)\)+(1/2)(Subscript[W, a  b  ]^(  p  q))(Subscript[R, pq]).

WdUdU = Raise[Wdddd, 2, 4];

cd1 = covDiv[WdUdU, {2, {4}}]; // Timing

BachTdd = 
          FacSimp[covDiv[cd1, {2}] + 
              multiDot[WdUdU, Rdd, {2, 1}, {4, 2}]/2]; // Timing

zeroQ[BachTdd]

•  This tensor vanishes when the space is conformal to an Einstein space, i.e., when M[t] is a constant.

zeroQ[FacSimp[BachTdd /. FuncRepRules[M[t], M0]]]

•  False because BachTensor contains up to the 4th derivatives of M[t]

zeroQ[FacSimp[BachTdd /. FuncRepRules[M[t], M0, 4]]]

\(
\)

• Bell-Robinson Tensor definition \(\(1\/4\) \((W\_\(\(a\)\(\ \ \)\(b\)\(\ \ \)\)\%\(\(\ \ \)\(p\ \ q\)\)\)\)(Subscript[W, cpdq])+*(Subscript[W, a  b  ]^(  p  q))*(Subscript[W, cpdq])), where *(Subscript[W, abcd])=(1/2)(Subscript[η, ab  ]^(    pq)) (Subscript[W, pqcd]). It is symmetric and traceless. It is also divergenceless in an Einstein space.

starWdddd = 
        multiDot[eta[3, 4], Wdddd/2, {3, 1}, {4, 2}];

starWdUdU = Raise[starWdddd, 2, 4];

BRTdddd = 
        FacSimp[multiDot[WdUdU, Wdddd, {2, 2}, {4, 4}]/4 + 
            multiDot[starWdUdU, starWdddd, {2, 2}, {4, 4}]/4];

(** Symmetric **)
zeroQ[BRTdddd - Transpose[BRTdddd, {1, 3, 2, 4}]]

zeroQ[BRTdddd - Transpose[BRTdddd, {4, 2, 3, 1}]]

•  Raise indices to be able to form trace / divergence.

BRTUUdd = Raise[BRTdddd, 1, 2];

(** Traceless **)
zeroQ[Contract[BRTUUdd, {1, 3}]]

(** Divergenceless if Einstein space **)
\(\((divBRT = covDiv[BRTUUdd, {1, {2}}]; 
	zeroQ[divBRT])\) // 
        Timing\)

zeroQ[divBRT /. FuncRepRules[M[t], M0, 4]]

zeroQ[divBRT /. FuncRepRules[Ω[t, r], 1]]

zeroQ[
      divBRT /. Join[FuncRepRules[Ω[t, r], 1], 
          FuncRepRules[M[t], M0, 4]]]